| Title:
|
Exponential separability is preserved by some products (English) |
| Author:
|
Tkachuk, Vladimir V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
385-395 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma$-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $|Y|\leq 2^{\omega_1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $|Y|\leq 2^{\mathfrak{c}} $ is scattered, then $X$ is exponentially separable. A Lindelöf $\Sigma$-space $X$ must be exponentially separable provided that every $Y\subset X$ with $|Y|\leq {\mathfrak{c}}$ is scattered. Under the Luzin axiom ($2^{\omega_1}>{\mathfrak{c}} $) we characterize weak exponential separability of $C_p(X,[0,1])$ for any metrizable space $X$. Our results solve several published open questions. (English) |
| Keyword:
|
Lindelöf space |
| Keyword:
|
scattered space |
| Keyword:
|
$\sigma$-product |
| Keyword:
|
function space |
| Keyword:
|
$P$-space |
| Keyword:
|
exponentially separable space |
| Keyword:
|
product |
| Keyword:
|
functionally countable space |
| Keyword:
|
weakly exponentially separable space |
| MSC:
|
54C35 |
| MSC:
|
54D65 |
| MSC:
|
54G10 |
| MSC:
|
54G12 |
| idZBL:
|
Zbl 07655808 |
| idMR:
|
MR4542797 |
| DOI:
|
10.14712/1213-7243.2022.021 |
| . |
| Date available:
|
2023-02-13T11:08:45Z |
| Last updated:
|
2024-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151541 |
| . |
| Reference:
|
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